2024高三·全国·专题练习
1 . 十七世纪法国数学家、被誉为业余数学家之王的皮埃尔·德·费马提出的一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小.”它的答案是:“当三角形的三个角均小于
时,所求的点为三角形的正等角中心,即该点与三角形的三个顶点的连线两两成角
;当三角形有一内角大于或等于
时,所求点为三角形最大内角的顶点.”在费马问题中所求的点称为费马点. 试用以上知识解决下面问题:已知
的内角
所对的边分别为
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40ec9cff8627e76b61e6474e57d7a7ef.png)
(1)求
;
(2)若
,设点
为
的费马点,求
;
(3)设点
为
的费马点,
,求实数
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40ec9cff8627e76b61e6474e57d7a7ef.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44ac38c5cc951497a4a37778b191bcce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b8f8a1e38db0e55b9b1934569b24e74.png)
(3)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b01862dfc85d45102a1343c36cb6dfe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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2 . 定义 如果函数
和
的图像上分别存在点
和
关于
轴对称,则称函数
和
具有
关系.
(1)若
,试判断函数
和
是否具有
关系;
(2)若函数
和
不具有
关系,求实数
的取值范围;
(3)若函数
和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c04f47dc0f7298b89b4d9f6195686f85.png)
在区间
上具有
关系,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2faa63899873813748f6a28b8a92e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2faa63899873813748f6a28b8a92e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166703f700475d6bcd4b8ee7c71f2c7f.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c7c95672da462203194363bb292dbcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2faa63899873813748f6a28b8a92e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166703f700475d6bcd4b8ee7c71f2c7f.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b524386c37301f3cfad5f28ef7883ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c95bc37ed65d74fba895bf21c4db925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166703f700475d6bcd4b8ee7c71f2c7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545ac68beedab0a5490f97c88437a317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c04f47dc0f7298b89b4d9f6195686f85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e5b56723fc6f8297356195e084fd9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b23efa380a58dc9663ba06dc249ce721.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166703f700475d6bcd4b8ee7c71f2c7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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解题方法
3 . 已知函数的定义域为
,且
,
,则( )
A.![]() | B.![]() ![]() |
C.![]() | D.![]() ![]() |
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解题方法
4 . 正实数x,y满足:存在
和
,使得
,
,
,则
的最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe3105e48ecbd0167a1ebc9480d4b6a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f0da7936f5bf3c07a3122e9fd1f0ca8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/946a7b7cec9439408d5d3c3a32b15d51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a953ac738380029ff0143c4a4596c62f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2d4e0ff2f771eb4657193ec1afba3f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88584cf1df43e28d03592c7998b1653.png)
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解题方法
5 . 在
中,
.则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f9fabbbe61a759e52ec975215e2e7c.png)
______ (用弧度制表示),若
为
的中点,且
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f86271ee2b1858cf3f54d57d8473dad.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f29c36db60dd12810ad2e90989d422bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f9fabbbe61a759e52ec975215e2e7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89cae28c97255a8d5640377f39be2ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f86271ee2b1858cf3f54d57d8473dad.png)
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6 . 十七世纪法国数学家、被誉为业余数学家之王的皮埃尔・德・费马提出的一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小”,意大利数学家托里拆利给出了解答,当
的三个内角均小于
时,使得
的点
即为费马点;当
有一个内角大于或等于
时,最大内角的顶点为费马点.已知
,
,
分别是
三个内角
,
,
的对边,且
,若点
为
的费马点,
,则实数
的取值范围为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e8036a881da6a4eef036529028a11d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e3ff728c68ba198cc1d7dcd12b2cfed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73669a238770ba2989e71ec2d1468738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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解题方法
7 . 固定项链的两端,在重力的作用下项链所形成的曲线是悬链线
年,莱布尼茨等得出悬链线的方程为
,其中
为参数.当
时,该表达式就是双曲余弦函数,记为
,悬链线的原理常运用于悬索桥、架空电缆、双曲拱桥、拱坝等工程.已知三角函数满足性质:①导数:
;②二倍角公式:
;③平方关系:
.定义双曲正弦函数为
.
(1)写出
,
具有的类似于题中①、②、③的一个性质,并证明该性质;
(2)任意
,恒有
成立,求实数
的取值范围;
(3)正项数列
满足
,
,是否存在实数
,使得
?若存在,求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46a7e0115ce78639910150e39fdbdb0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f07f8015f0a035e80a166092be0b7318.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a7c1d3681898e25187a896aeb0c8c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf8bce35b539fdf365e9089750d4d152.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32eac4b7f177c041219fab18de973c5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9af7ca3fcd9a43d520ed650b80ef2dad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0718c04bdf70989bcc90b902671a692.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a53b8e0108664bf39aa302570457199a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6e7c627427318b62d977ff7a86c2cb5.png)
(2)任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a68fd5f6e28316a932db1494deac24b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)正项数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c19bf566cd9dd81916f53ed33248197c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f816db73b759d7de72b0bd43ba39f46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ecf3a1fecf89a37a677393d0bfe27b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/805dabba8d859d870a1dfaaa9d97de41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2024-06-02更新
|
446次组卷
|
2卷引用:福建省安溪第八中学2024届高三下学期5月份质量检测数学试题
2024·全国·模拟预测
8 . 已知函数
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5e0470a710a28b7ef5eb4b0777652a1.png)
A.![]() ![]() |
B.若![]() ![]() ![]() |
C.若![]() ![]() ![]() ![]() |
D.函数![]() ![]() |
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9 . 满足
且互不相似的
的个数为______ 个.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae512cb71b62247b6edb92ee701a4d61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2024·全国·模拟预测
解题方法
10 . 在
中,角
,
,
所对的边分别为
,
,
是边
上一点,且
,
,若
为钝角,则当
最小时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ec5d678ec42846e1d28301e3bfd4be.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d7bce5b3862e12e5c7d206c35052471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bc47c990c2477f433951c5cc37a9d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf9ad150cb1e4cd8977d4cc3d99be17c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a72dfbf0138a611174c36ce077e0c47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/451be539a00c5a0fa9e00fc9bf4c39fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ec5d678ec42846e1d28301e3bfd4be.png)
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